| Exam Board | Edexcel |
|---|---|
| Module | FD1 AS (Further Decision 1 AS) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.5 This is a standard graphical linear programming problem where the feasible region is already drawn and students must find the minimum of an objective function by testing corner points. While it requires systematic evaluation of vertices and understanding of optimization, it follows a routine algorithmic procedure taught in Decision Maths with no novel problem-solving required. |
| Spec | 7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Objective line drawn or at least two vertices tested | M1 | Selecting appropriate mathematical process – drawing objective line with correct gradient or reciprocal gradient, or testing at least two vertices in C |
| For solving \(y = 4x\) and \(8x + 7y = 560\) to find exact co-ordinate of optimal point, must reach either \(x =\) or \(y =\) | M1 | Solving simultaneous equations |
| \(x = 15\frac{5}{9}\) and \(y = 62\frac{2}{9}\) | A1 | cao |
| Finding at least two points with integer co-ordinates from \((15 \pm 1,\ 63 \pm 2)\) | M1 | Recognition that outcome is non-integer; testing two points with integer co-ordinates in at least one of \(y \geq 4x\) and \(8x + 7y \geq 560\) |
| Testing at least two points with integer co-ordinates | M1 | Testing at least two integer solutions in \(y \geq 4x\) or \(8x + 7y \geq 560\) and C |
| \(x = 15\) and \(y = 63\) | A1 | cao – deducing which integer solution is both valid and optimal |
| So the teacher should buy 15 pens and 63 pencils | A1ft | Interpreting solution in context – gives integer values for \(x\) and \(y\) in context of pens and pencils |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Objective line drawn or at least two vertices tested | M1 | Selecting appropriate mathematical process – drawing objective line with correct gradient or reciprocal gradient, or testing at least two vertices in C |
| For solving $y = 4x$ and $8x + 7y = 560$ to find exact co-ordinate of optimal point, must reach either $x =$ or $y =$ | M1 | Solving simultaneous equations |
| $x = 15\frac{5}{9}$ and $y = 62\frac{2}{9}$ | A1 | cao |
| Finding at least two points with integer co-ordinates from $(15 \pm 1,\ 63 \pm 2)$ | M1 | Recognition that outcome is non-integer; testing two points with integer co-ordinates in at least one of $y \geq 4x$ and $8x + 7y \geq 560$ |
| Testing at least two points with integer co-ordinates | M1 | Testing at least two integer solutions in $y \geq 4x$ or $8x + 7y \geq 560$ and C |
| $x = 15$ and $y = 63$ | A1 | cao – deducing which integer solution is both valid and optimal |
| So the teacher should buy 15 pens and 63 pencils | A1ft | Interpreting solution in context – gives integer values for $x$ and $y$ in context of pens and pencils |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{1e2c1dc4-3724-4bba-961c-1c2ae7e649c4-3_1463_1194_239_440}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A teacher buys pens and pencils. The number of pens, $x$, and the number of pencils, $y$, that he buys can be represented by a linear programming problem as shown in Figure 2, which models the following constraints:
$$\begin{aligned}
8 x + 3 y & \leqslant 480 \\
8 x + 7 y & \geqslant 560 \\
y & \geqslant 4 x \\
x , y & \geqslant 0
\end{aligned}$$
The total cost, in pence, of buying the pens and pencils is given by
$$C = 12 x + 15 y$$
Determine the number of pens and the number of pencils which should be bought in order to minimise the total cost. You should make your method and working clear.\\
\hfill \mbox{\textit{Edexcel FD1 AS Q2 [7]}}